A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

“…The reason we can use this method here is that a Lions' type lemma also holds in our case. Such kind of lemma has been used as a powerful tool in [20] for compact Riemannian surface without boundary and in [12] for bounded smooth domains in R 4 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A class of Adams–Fontana type inequalities and related functionals on manifolds

Yang

Zhao

2009

Nonlinear Differ. Equ. Appl.

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A class of Adams-Fontana type inequalities are established on compact Riemannian manifolds without boundary via the Young inequality together with the usual Adams-Fontana inequality (Comment Math Helv 68:415-454, 1993). As an application, a sequence of functionals are defined on manifolds, a sufficient condition on which the Palais-Smale condition holds is given and the existence of critical points of the functionals is also considered in the spirit of Adimurthi (Ann Scuola Norm Sup Pisa Cl Sci 17:393-413, 1990) and Adimurthi and Sandeep (Nonlinear Differ Equ Appl 13:585-603, 2007).

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“…This contradicts (18) when ǫ is sufficiently small and implies that u ǫ is uniformly bounded in B 1 . Then applying elliptic estimates to (17), we have that u ǫ converges to an extremal function u * in C 1 (B 1 ), as desired.…”

Section: Proof Of Corollarymentioning

confidence: 95%

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

Yang

2018

Arch. Math.

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In this short note, we generalized an energy estimate due to Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to reprove existence of extremals for Trudinger-Moser inequalities of Adimurthi-Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.

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A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

Abstract: Abstract. In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved.

Cited by 89 publications

References 16 publications

A sharp form of Moser–Trudinger inequality in high dimension

A sharp form of Moser–Trudinger inequality in high dimension

A class of Adams–Fontana type inequalities and related functionals on manifolds

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

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